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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Complex Roots</dfn> The characteristic equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r^n+a_1 r^{n-1}+a_{n-1} r+a_n=0.
\end{equation*}
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<p class="continuation">It has total <span class="process-math">\(n\)</span> roots. Suppose that there is a pair of complex conjugate roots. <span class="process-math">\(\lambda+i \mu\)</span> and <span class="process-math">\(\lambda-i\mu\text{.}\)</span> Then we have solutions</p>
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\begin{equation*}
e^{(\lambda+i\mu) x}, e^{(\lambda-i\mu) x},
\end{equation*}
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<p class="continuation">which can be replaced by</p>
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\begin{equation*}
e^{\lambda x} \cos \mu x, \quad e^{\lambda x} \sin \mu x.
\end{equation*}
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<span class="incontext"><a href="sec4_2.html#p-158" class="internal">in-context</a></span>
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